It is important to notice the difference between the model in Sec.  and those in Sec. . In both there is a mixing in the vector kinetic terms, but as in the former, the kinetic terms are diagonalized before the spontaneous symmetry breaking (SSB). On the other hand, in the model of Sec.  [1]}, [2]}, the diagonalization of the kinetic mixing is done
after the SSB, \(M^2_{FLN}=K^TM^2_{KN}K\)  where \(K\)  is the \(3\times 3\)  extension of the matrix defined in Eq. (REF ) and \(M^2_{KN}\)  is the matrix in Eq. (REF ). The result is the matrix of Eq. (A2) of Ref. [2]} that has determinant zero.
In fact, if in the model of Sec.  the kinetics mixing had occurred before the SSM, the mass matrix of the neutral vector bosons would be
\(M^2_{FLN}=\frac{g^2v^2}{4c^2_W}\left(\begin{array}{ccc}\xi ^2 s^2_W+r^2_1 & \xi s^2_W+r_1r_2 & -\xi c_Ws_W \\\xi s^2_W+ r_1r_2& s^2_W+r^2_2 & -c_Ws_W \\-\xi c_Ws_W & -c_Ws_W & c^2_W\end{array}\right),\) 
